How can I make a flat, spiralled surface?
In Object Mode add Shift-A a circle (Mesh/Circle)
 
Change to Edit Mode Tab make sure you're in vertex select mode.
Extrude and Scale E,S,0
Change to Face select (see image below)
Click a random face, select all A
From the menu pick Select/ Checker Deselect





Create another material and assign it to the selection
Add loop cuts Ctrl-R RMB

With the inner faces select you're there
Set proprtional editing O and rotate R



You might want to select the outer edges and extrude them to add the on colored border as in the reference image.

Result with some tweaking: